Is this a joke? This is an old theorem.
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Qiaochu YuanOct 26 '09 at 23:17

No, not a joke, I'm pretty bad in standard number theory facts and I confused it with a bunch of other primes-related hard statements. I am very red-faced now though :)
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Ilya NikokoshevOct 26 '09 at 23:20

Well, this was of course quite a stupid question of me because most part of analytic number theory would make no sense if such a bad approximation to pi(x) was not known...
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Ilya NikokoshevOct 26 '09 at 23:22

Bertrand's Postulate is not a consequence of the Prime Number Theorem, so it doesn't follow that if BP was unproven then even a bad approximation of pi(x) would be beyond reach. What I'm trying to say is, the theorem and the proof are well-known but it's not quite such a stupid question :-)
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Alon AmitOct 27 '09 at 4:45

There are various proofs of Bertrand's postulate. There is quite an easy one available if one treats it together with the proof of the usual (double) Chebyshev bound as a unit. One optimizes the proof of the Chebyshev bound for subsequently proving Bertrand's Postulate by the method of Ramanujan (getting rid of the appeal to Stirling's formula at the same time).

The history of Bertrand's Postulate is set forth in The Development of Prime Number Theory by Wladyslaw Narkiewicz.

A comment to the comment by Michael Lugo. The Prime Number Theorem is considerably harder to prove than Bertrand's Postulate, and getting the PNT in the form of good explicit inequalities is hard work on top of that (such inequalities exist, and are useful for some purposes).

Note that we can also give a conditional proof of Bertrand's Postulate assuming the veracity of Goldbach's Conjecture. This was the subject matter of a short note that appeared in the sixth issue of volume #112 of the Monthly.

Also, it has to be noted that the reference given by Michael Lugo doesn't contain the original proof of Erdős. The original one is to be found here.

Thanks for the pointer to the original proof. The reference I gave is where I first saw the proof. (And the original proof was published in German; my German is not so good.)
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Michael LugoNov 8 '09 at 18:29

One remark to relate Bertrand's postulate to the prime number theorem: Chebyshev's work was related to bounding ratios of factorials--in particular $\frac{2n!}{n!n!}.$

His later proof that $C\frac{x}{\log x} < \pi(x) < D\frac{x}{\log x}$ made use of other ratios (in this case $\frac{30n!n!}{15n!5n!3n!}$). In theory one could try and improve the numbers used in the ratio to asymptotically prove the prime number theorem. Jonathan Bober (among others) have worked on this. He has catalogued many different combinations of ratios of factorials (this also ends up tying into G and E functions....but I'm already out of my depth of what I'm capable of explaining).